From 8ef2849b678e1e76b1caf4c6a841690d9d34f1d1 Mon Sep 17 00:00:00 2001 From: Rob Lewis Date: Mon, 23 Feb 2015 13:05:24 -0500 Subject: [PATCH] feat(library/algebra/fields): prove more theorems about division rings --- library/algebra/field.lean | 86 ++++++++++++++++++++++---------------- 1 file changed, 51 insertions(+), 35 deletions(-) diff --git a/library/algebra/field.lean b/library/algebra/field.lean index e3702d019..9f661ab5c 100644 --- a/library/algebra/field.lean +++ b/library/algebra/field.lean @@ -8,7 +8,7 @@ Authors: Robert Lewis Structures with multiplicative and additive components, including division rings and fields. The development is modeled after Isabelle's library. -/ - +---------------------------------------------------------------------------------------------------- import logic.eq logic.connectives data.unit data.sigma data.prod import algebra.function algebra.binary algebra.group algebra.ring open eq eq.ops @@ -21,8 +21,6 @@ structure division_ring [class] (A : Type) extends ring A, has_inv A := (mul_inv_cancel : ∀{a}, a ≠ zero → mul a (inv a) = one) (inv_mul_cancel : ∀{a}, a ≠ zero → mul (inv a) a = one) --- theorem div_is_mul [s : division_ring A] {a b : A} : a / b = a * b⁻¹ := rfl - section division_ring variables [s : division_ring A] {a b c : A} include s @@ -51,11 +49,7 @@ section division_ring theorem div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H - theorem mul_div_assoc (Hc : c ≠ 0) : (a * b) / c = a * (b / c) := - eq.symm (calc - a * (b / c) = a * (b * c⁻¹) : rfl - ... = (a * b) * c⁻¹ : mul.assoc - ... = (a * b) / c : rfl) + theorem mul_div_assoc : (a * b) / c = a * (b / c) := !mul.assoc theorem one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 := assume H2 : 1 / a = 0, @@ -73,13 +67,12 @@ section division_ring theorem div_one : a / 1 = a := calc - a / 1 = /- a * 1⁻¹ : rfl - ... = -/ a * 1 : inv_one_is_one + a / 1 = a * 1 : inv_one_is_one ... = a : mul_one - -- note: integral domain has a "mul_ne_zero". When we get to "field", show it is an - -- instance of an integral domain, so we can use that theorem. - -- check @mul_ne_zero + theorem zero_div : 0 / a = 0 := !zero_mul + + -- note: integral domain has a "mul_ne_zero". Discrete fields are int domains. theorem mul_ne_zero' (Ha : a ≠ 0) (Hb : b ≠ 0) : a * b ≠ 0 := assume H : a * b = 0, have C1 : a = 0, from (calc @@ -90,6 +83,7 @@ section division_ring ... = 0 : zero_mul), absurd C1 Ha + -- this belongs in ring? theorem mul_ne_zero_imp_ne_zero (H : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 := have Ha : a ≠ 0, from (assume Ha1 : a = 0, @@ -180,6 +174,25 @@ section division_ring ... = (1 / a) * (-1) : one_div_neg_one_eq_neg_one ... = - (1 / a) : mul_neg_one_eq_neg + theorem div_neg_eq_neg_div (Ha : a ≠ 0) : b / (- a) = - (b / a) := + calc + b / (- a) = b * (1 / (- a)) : inv_eq_one_div + ... = b * -(1 / a) : one_div_neg_eq_neg_one_div Ha + ... = -(b * (1 / a)) : neg_mul_eq_mul_neg + ... = - (b * a⁻¹) : inv_eq_one_div + + theorem neg_div (Ha : a ≠ 0) : (-b) / a = - (b / a) := + calc + (-b) / a = (-1 * b) / a : neg_eq_neg_one_mul + ... = (-1) * (b / a) : mul_div_assoc + ... = - (b / a) : neg_eq_neg_one_mul + + theorem neg_div_neg_eq_div (Hb : b ≠ 0) : (-a) / (-b) = a / b := + calc + (-a) / (-b) = - ((-a) / b) : div_neg_eq_neg_div Hb + ... = - -(a / b) : neg_div Hb + ... = a / b : neg_neg + theorem div_div (H : a ≠ 0) : 1 / (1 / a) = a := symm (eq_one_div_of_mul_eq_one_left (mul_one_div_cancel H)) @@ -201,24 +214,17 @@ section division_ring theorem mul_div_cancel (Hb : b ≠ 0) : a * b / b = a := calc - (a * b) / b = a * b * b⁻¹ : rfl - ... = a * (b * b⁻¹) : mul.assoc + (a * b) / b = a * (b * b⁻¹) : mul.assoc ... = a * 1 : mul_inv_cancel Hb ... = a : mul_one theorem div_mul_cancel (Hb : b ≠ 0) : a / b * b = a := calc - (a / b) * b = (a * b⁻¹) * b : rfl - ... = a * (b⁻¹ * b) : mul.assoc + (a / b) * b = a * (b⁻¹ * b) : mul.assoc ... = a * 1 : inv_mul_cancel Hb ... = a : mul_one - theorem div_add_div_same (Hc : c ≠ 0) : a / c + b / c = (a + b) / c := - calc - (a / c) + (b / c) = (a * c⁻¹) + (b / c) : rfl - ... = a * c⁻¹ + b * c⁻¹ : rfl - ... = (a + b) * c⁻¹ : right_distrib - ... = (a + b) / c : rfl + theorem div_add_div_same : a / c + b / c = (a + b) / c := !right_distrib⁻¹ theorem inv_mul_add_mul_inv_eq_inv_add_inv (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (a + b) * (1 / b) = 1 / a + 1 / b := @@ -249,6 +255,21 @@ section division_ring a / b = b / b : H2 ... = 1 : div_self Hb) + theorem eq_div_iff_mul_eq (Hc : c ≠ 0) : a = b / c ↔ a * c = b := + iff.intro + (assume H : a = b / c, calc + a * c = b / c * c : H + ... = b : div_mul_cancel Hc) + (assume H : a * c = b, symm (calc + b / c = a * c / c : H + ... = a : mul_div_cancel Hc)) + + theorem add_div_eq_mul_add_div (Hc : c ≠ 0) : a + b / c = (a * c + b) / c := + have H : (a + b / c) * c = a * c + b, from calc + (a + b / c) * c = a * c + (b / c) * c : right_distrib + ... = a * c + b : div_mul_cancel Hc, + (iff.elim_right (eq_div_iff_mul_eq Hc)) H + end division_ring structure field [class] (A : Type) extends division_ring A, comm_ring A @@ -256,6 +277,7 @@ structure field [class] (A : Type) extends division_ring A, comm_ring A section field variables [s : field A] {a b c d: A} include s + local attribute divide [reducible] -- I think of "div_cancel" has being something like a * b / b = a or a / b * b = a. The name -- I chose is clunky, but it has the right prefix @@ -268,8 +290,7 @@ section field ... = a * ((1 / a) * (1 / b)) :inv_eq_one_div ... = a * (1 / (b * a)) : one_div_mul_one_div Ha Hb ... = a * (1 / (a * b)) : mul.comm - ... = a * (a * b)⁻¹ : inv_eq_one_div - ... = a / (a * b) : rfl) + ... = a * (a * b)⁻¹ : inv_eq_one_div) theorem div_mul_left (Ha : a ≠ 0) (H : a * b ≠ 0) : b / (a * b) = 1 / a := have H1 : b * a ≠ 0, from mul_ne_zero_comm H, @@ -290,21 +311,17 @@ section field theorem one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) := have H : a * b ≠ 0, from (mul_ne_zero' Ha Hb), symm (calc - (a + b) / (a * b)/- = (a + b) * (a * b)⁻¹ : rfl - ...-/ = a * (a * b)⁻¹ + b * (a * b)⁻¹ : right_distrib + (a + b) / (a * b) = a * (a * b)⁻¹ + b * (a * b)⁻¹ : right_distrib ... = a / (a * b) + b * (a * b)⁻¹ : rfl ... = 1 / b + b * (a * b)⁻¹ : div_mul_right Hb H - ... = 1 / b + b / (a * b) : rfl ... = 1 / b + 1 / a : div_mul_left Ha H ... = 1 / a + 1 / b : add.comm) theorem div_mul_div (Hb : b ≠ 0) (Hd : d ≠ 0) : (a / b) * (c / d) = (a * c) / (b * d) := calc - (a / b) * (c / d) = (a * b⁻¹) * (c / d) : rfl - ... = (a * b⁻¹) * (c * d⁻¹) : rfl + (a / b) * (c / d) = (a * b⁻¹) * (c * d⁻¹) : rfl ... = (a * c) * (d⁻¹ * b⁻¹) : by rewrite [2 mul.assoc, (mul.comm b⁻¹), mul.assoc] ... = (a * c) * (b * d)⁻¹ : mul_inv Hd Hb - ... = (a * c) / (b * d) : rfl theorem mul_div_mul_left (Hb : b ≠ 0) (Hc : c ≠ 0) : (c * a) / (c * b) = a / b := have H : c * b ≠ 0, from mul_ne_zero' Hc Hb, @@ -322,8 +339,7 @@ section field theorem div_mul_eq_mul_div (Hc : c ≠ 0) : (b / c) * a = (b * a) / c := calc (b / c) * a = (b * c⁻¹) * a : rfl - ... = (b * a) * c⁻¹ : by rewrite [mul.assoc, (mul.comm c⁻¹), -mul.assoc ] - ... = (b * a) / c : rfl + ... = (b * a) * c⁻¹ : by rewrite [mul.assoc, (mul.comm c⁻¹), -mul.assoc] -- this one is odd -- I am not sure what to call it, but again, the prefix is right theorem div_mul_eq_mul_div_comm (Hc : c ≠ 0) : (b / c) * a = b * (a / c) := @@ -339,14 +355,14 @@ section field calc a / b + c / d = (a * d) / (b * d) + c / d : mul_div_mul_right Hb Hd ... = (a * d) / (b * d) + (b * c) / (b * d) : mul_div_mul_left Hd Hb - ... = ((a * d) + (b * c)) / (b * d) : div_add_div_same H + ... = ((a * d) + (b * c)) / (b * d) : div_add_div_same theorem div_sub_div (Hb : b ≠ 0) (Hd : d ≠ 0) : (a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) := calc (a / b) - (c / d) = (a / b) + -1 * (c / d) : neg_eq_neg_one_mul - ... = (a / b) + ((-1 * c) / d) : mul_div_assoc Hd + ... = (a / b) + ((-1 * c) / d) : mul_div_assoc ... = ((a * d) + (b * (-1 * c))) / (b * d) : div_add_div Hb Hd ... = ((a * d) + -1 * (b * c)) / (b * d) : by rewrite [-mul.assoc, (mul.comm b), mul.assoc] ... = ((a * d) + -(b * c)) / (b * d) : neg_eq_neg_one_mul